Higher order shock structure for a class of generalized Burgers' equations

A class of generalized Burgers' equations is considered in which the nonlinearity is of arbitrary form whereas the dissipation is linear with small coefficient k. The solution is shown to develop shocks and the structure of the solution both within the shock and in the outer region is obtained accurately to order k by means of matched asymptotic expansions. It is shown that to lowest order the shock position can be determined by an extended version of Whitham's equal-areas rule whereas to order k a general explicit expression for Lighthill's 'displacement due to diffusion' is derived.